Most of the registration process on websites, ask you for the country name from drop down. My question is , should we design our db like this:
Table 1: Country (countryID, countryName)
Table 2: Registration(Name, email ... Country(Id/Name) .

So, it should be countryId in the registration table or countryName ?

Which is the best approach ?

That really depends on if you will be making frequent queries to just the countries. If you seperate them then your db wouldn't have to filter through all the other stuff so in essence it would speed things up a bit, however if you will only be pulling the country info in correlation to someones profile or in other words with every thing else, then if the countries table is seperate you would either have to do two queries or use a join which starts to get confusing.

so mean, it should depend on the application requirements?

Exactly, there is never a one size fits all database design.

Using the country ID and a join makes for a better DB design.

It's a process called "normalisation". Where the DB designer reduces the amount of redundant (duplicated) data in the tables.

I totally agree with chrishirst.

I usually use 3 tables for that.
There is a typical mysql schema.

One table with a country list:

create table country(
  iso char 3,
  name varchar(255),
  
  primary key (iso)
);

One table with my user infos, but no direct links with the country table

create table user(
  id number not null,
  username varchar(50) not null,
  password varchar(200) not null,

  primary key (id)
);

And finally one last table who link the users and the country:

create table countryLink(
  userId number not null,
  iso char(3) not null,

  primary key (userId,iso)
);